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D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"075111d0f4363c9ecb565dad58e644c8393c643484340d67a7df931a8e80fcbd","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-a6f3b35778192ffb","role":"sequence","name":"Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:  1;\n  1:  2,   1;\n  2:  4,   5,    1;\n  3:  7,  18,    9,     1;\n  4: 11,  48,   49,    14,     1;\n  5: 16, 105,  180,   105,    20,     1;\n  6: 22, 201,  513,   501,   195,    27,    1;\n  7: 29, 350, 1232,  1785,  1162,   329,   35,   1;\n  8: 37, 568, 2618,  5212,  5027,  2380,  518,  44,  1;\n  9: 46, 873, 5076, 13182, 17442, 12231, 4452, 774, 54, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^3-2*n^2+3*n-2-(5*n^3-18*n^2+17*n-4)*x+(4*n^3-18*n^2+32*n-18)*x^2+(8*n^2-22*n+14)*x^3+(4*n-4)*x^4)*P(n,x) = (n^3+n-2-(5*n^3-16*n^2+3*n+8)*x+(n^3-2*n^2+31*n-30)*x^2+(12*n^3-14*n^2-6*n+8)*x^3+(24*n^2-38*n+14)*x^4+(12*n-12)*x^5)*P(n-1,x) + ((2*n^2-2*n-4)*x-(6*n^2+2*n+8)*x^2-(6*n^2+18*n+4)*x^3-(8*n^2+16*n+8)*x^4-(16*n+4)*x^5-8*x^6)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((n^3 + n - 2 - (5*n^3 - 16*n^2 + 3*n + 8)*x + (n^3 - 2*n^2 + 31*n - 30)*x^2 + (12*n^3 - 14*n^2 - 6*n + 8)*x^3 + (24*n^2 - 38*n + 14)*x^4 + (12*n - 12)*x^5)*P[n - 1, x] + ((2*n^2 - 2*n - 4)*x - (6*n^2 + 2*n + 8)*x^2 - (6*n^2 + 18*n + 4)*x^3 - (8*n^2 + 16*n + 8)*x^4 - (16*n + 4)*x^5 - 8*x^6)*D[P[n - 1, x], x])/(n^3 - 2*n^2 + 3*n - 2 - (5*n^3 - 18*n^2 + 17*n - 4)*x + (4*n^3 - 18*n^2 + 32*n - 18)*x^2 + (8*n^2 - 22*n + 14)*x^3 + (4*n - 4)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = ((n^3 + n - 2 - (5*n^3 - 16*n^2 + 3*n + 8)*x + (n^3 - 2*n^2 + 31*n - 30)*x^2 + (12*n^3 - 14*n^2 - 6*n + 8)*x^3 + (24*n^2 - 38*n + 14)*x^4 + (12*n - 12)*x^5)*v[i - 1] + ((2*n^2 - 2*n - 4)*x - (6*n^2 + 2*n + 8)*x^2 - (6*n^2 + 18*n + 4)*x^3 - (8*n^2 + 16*n + 8)*x^4 - (16*n + 4)*x^5 - 8*x^6)*deriv(v[i - 1], x))/(n^3 - 2*n^2 + 3*n - 2 - (5*n^3 - 18*n^2 + 17*n - 4)*x + (4*n^3 - 18*n^2 + 32*n - 18)*x^2 + (8*n^2 - 22*n + 14)*x^3 + (4*n - 4)*x^4)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-a6f3b35778192ffb","steps":["D+1"],"edges":["e-f5467c80bb90d43f"],"text":"A056241 \u2192 D+1"},"headline":"A056241 \u2192 D+1","chain":null}